We have shown how to use the Laplace
transform to solve linear differential equations. Familiar functions
that arise in solutions to differential equations
are and . Theorem
12.15 (the first shifting theorem) shows how their transforms are
related to those of and by
shifting the variable s in . A
companion result, called the second shifting theorem, Theorem 12.16,
shows how the transform of
can be obtained by multiplying
by . Loosely
speaking, these results show that multiplication of
by corresponds
to shifting , and
that shifting
corresponds to multiplication of the transform
by .

Theorem 12.15 (Shifting
the
Variable s). If is
the Laplace transform of , then